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In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister () for 3-manifolds and generalized to higher dimensions by and . Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by as an analytic analogue of Reidemeister torsion. and proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds. Reidemeister torsion was the first invariant in algebraic topology that could distinguish between spaces which are homotopy equivalent but not homeomorphic, and can thus be seen as the birth of geometric topology as a distinct field. It can be used to classify lens spaces. Reidemeister torsion is closely related to Whitehead torsion; see . For later work on torsion see the books , . And it had given one of important motivation to arithmetic topology. ==Definition of analytic torsion== If ''M'' is a Riemannian manifold and ''E'' a vector bundle over ''M'', then there is a Laplacian operator acting on the ''i''-forms with values in ''E''. If the eigenvalues on ''i''-forms are λ''j'' then the zeta function ζ''i'' is defined to be : for ''s'' large, and this is extended to all complex ''s'' by analytic continuation. The zeta regularized determinant of the Laplacian acting on ''i''-forms is : which is formally the product of the positive eigenvalues of the laplacian acting on ''i''-forms. The analytic torsion ''T''(''M'',''E'') is defined to be : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「analytic torsion」の詳細全文を読む スポンサード リンク
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